Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

95% confidence interval

95 %CI¼exp½ 0 : 2214  1 : 96 ð 0 : 8558 ފ

¼ð 0 : 23 ; 6 : 68 Þ

Wald test

H 0 :b 3 ¼ 0

Z¼

0 : 2214

0 : 8558

¼ 0 : 259 ;P¼ 0 : 7958

Standard Logistic Regression
Model

Variable Coefficient Std Err

Wald

p-value

INTERCEPT 1.4362 0.6022 0.0171
BIRTHWGT 0.0005 0.0002 0.0051
GENDER 0.0453 0.2757 0.8694
DIARRHEA 0.7764 0.4538 0.0871

Responses within clusters assumed
independent

Also called the “naive” model

Odds ratio

ORdðDIARRHEA¼ 1 vs:DIARRHEA¼ 0 Þ

¼expð 0 : 7764 Þ¼ 2 : 17

95% confidence interval

95 %CI¼exp½ 0 : 7764  1 : 96 ð 0 : 4538 ފ

¼ð 0 : 89 ; 5 : 29 Þ

The 95% confidence interval is calculated

using the usual large-sample formula, yielding

a confidence interval of (0.23, 6.68).

We can test the null hypothesis that the beta

coefficient for DIARRHEA is equal to zero

using the Wald test, in which we divide the

parameter estimate by its standard error. For

the variable DIARRHEA, the Wald statistic

equals 0.259. The corresponding P-value is

0.7958, which indicates that there is not

enough evidence to reject the null hypothesis.

The output for the standard logistic regression

is presented for comparison. In this analysis,

each observation is assumed to be indepen-

dent. When there are several observations per

subject, as with these data, the term “naive

model” is often used to describe a model

that assumes independence when responses

within a cluster are likely to be correlated. For

the Infant Care Study example, there are 1,203

separate outcomes across the 136 infants.

Using this output, the estimated odds ratio

comparing symptoms of diarrhea vs. no diar-

rhea is 2.17 for the naive model.

The 95% confidence interval for this odds ratio

is calculated to be (0.89, 5.29).

496 14. Logistic Regression for Correlated Data: GEE